A strict implication calculus for compact Hausdorff spaces

Annals of Pure and Applied Logic 170 (2019) 102714

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Annals of Pure and Applied Logic

www.elsevier.com/locate/apal

A strict implication calculus for compact Hausdorff spaces

G. Bezhanishvili a, N. Bezhanishvili b,∗, T. Santoli c, Y. Venema b

a NMSU, Las Cruces, USAb ILLC, Universiteit van Amsterdam, The Netherlandsc Mathematical Institute, University of Oxford, UK

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 May 2017Received in revised form 20 January 2019Accepted 5 April 2019Available online 1 July 2019

MSC:03B4506E1506E2554E0554G05

Keywords:Compact Hausdorff spaceStone spaceExtremally disconnected spaceGleason coverBoolean algebra with operatorsde Vries algebraModal logicStrict implication

We introduce a simple modal calculus for compact Hausdorff spaces. The language of our system extends that of propositional logic with a strict implication connective, which, as shown in earlier work, algebraically corresponds to the notion of a subordination on Boolean algebras. Our base system is a strict implication calculus SIC, to which we associate a variety SIA of strict implication algebras. We also study the symmetric strict implication calculus S2IC, which is an extension of SIC, and prove that S2IC is strongly sound and complete with respect to de Vries algebras. By de Vries duality, this yields completeness of S2IC with respect to compact Hausdorff spaces. Since some of the defining axioms of de Vries algebras are Π2-sentences, we develop the corresponding theory of non-standard rules, which we term Π2-rules. We study the resulting inductive elementary classes of algebras, and give a general criterion of admissibility for Π2-rules. We also compare our approach to approaches in the literature that are related to our work.1

© 2019 Elsevier B.V. All rights reserved.

1. Introduction

Extending Stone’s seminal representation theorems for Boolean algebras [25] and distributive lattices [26], categorical dualities linking algebra and topology have been of fundamental importance in the development of the 20th century mathematics in general [22], and of logic and theoretical computer science in particular [18]. With algebras corresponding to the syntactic, deductive side of logical systems, and topological spaces

* Corresponding author.E-mail addresses: [email protected] (G. Bezhanishvili), [email protected] (N. Bezhanishvili), [email protected]

(T. Santoli), [email protected] (Y. Venema).1 The results presented in this paper were first reported in [24] (see also [5]).

https://doi.org/10.1016/j.apal.2019.06.0030168-0072/© 2019 Elsevier B.V. All rights reserved.

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to their semantics, Stone-type duality theory provides an elegant and useful mathematical framework for studying various properties of logical systems. In many particular cases, one sees natural specimens of logics, classes of algebras, and classes of topologies coming together. Out of a multitude of examples of such triples, we mention: (i) classical logic/Boolean algebras/Stone spaces [2]; (ii) intuitionistic logic/Heyting algebras/Esakia spaces [15]; and (iii) modal logic/modal algebras/topological Kripke frames [7].

Our aim is to add to the study of these ‘logic/algebra/topology’ triples by providing a simple logical calculus for reasoning about compact Hausdorff spaces—a widely studied class of spaces, properly containing the class of Stone spaces. We do this by generalizing the classical setting. Namely, we extend the classical propositional language with a new logical connective of strict implication, which admits a natural topological interpretation; and we design a calculus consisting of finitely many axioms and rules that is sound and complete with respect to a category of algebras which is dual to the category KHaus of compact Hausdorff spaces and continuous maps. Our framework is built on de Vries duality, which we will now discuss in more detail.

A subordination on a Boolean algebra B is a binary relation ≺ on B satisfying certain conditions (see Definition 2.1). Subordinations were introduced in [6]. They are in one-to-one correspondence with quasi-modal operators of [9] and pre-contact relations of [13]. Subordinations can be modeled dually by closed relations on Stone spaces. This leads to a duality between the category Sub of subordination algebras and the category StR of pairs (X, R) where X is a Stone space and R is a closed relation on X (see [6, Sec. 2.1]). Further conditions on ≺ characterize when the relation R is reflexive or symmetric. This yields the subcate-gories RSub and Con of Sub consisting of reflexive subordination algebras and contact algebras, respectively. As the name suggests, reflexive subordinations dually correspond to reflexive closed relations; while contact relations correspond to reflexive and symmetric closed relations.

Compingent algebras, introduced by de Vries [12], are obtained by adding two additional conditions to the definition of contact algebras. As was shown in [6, Lem. 6.3], they are dually characterized by irreducible equivalence relations (the definition is given in Section 2). Thus, the category Com of compingent algebras is dually equivalent to the subcategory of StR consisting of the pairs (X, R) where R is an irreducible equivalence relation. A de Vries algebra is a complete compingent algebra. Since complete Boolean algebras dually correspond to extremally disconnected Stone spaces, we conclude that the category DeV of de Vries algebras is dually equivalent to the subcategory of StR consisting of the pairs (X, R) where the Stone space X is extremally disconnected and R is an irreducible equivalence relation. Such pairs were called Gleason spaces in [6] since they are closely related to Gleason covers of compact Hausdorff spaces. A key result of [6, Thm. 6.13] is that this close correspondence between Gleason spaces and Gleason covers yields that the category Gle of Gleason spaces is equivalent to KHaus. Since Gle is dually equivalent to DeV, we arrive at de Vries duality: KHaus is dually equivalent to DeV.

It was pointed out in [6, Sec. 3] that subordinations on a Boolean algebra B can alternatively be de-scribed by binary operations on B called strict implications. In this paper we study the resulting variety of strict implication algebras. The study simplifies considerably if we work with the strict implications that correspond to reflexive subordinations. In Section 3 we prove that the resulting variety SIA of strict im-plication algebras is a discriminator variety, and we give its axiomatization. We also prove that SIA is a locally finite variety. In Section 4 we develop the corresponding strict implication calculus SIC, which is a modal logic with one binary modality that corresponds to the strict implication. Section 5 is devoted to the symmetric strict implication calculus S2IC which is an extension of SIC. The corresponding variety S2IA is the subvariety of SIA generated by the strict implication algebras that correspond to contact algebras. One of our main results is that S2IA is generated by the strict implication algebras that correspond to de Vries algebras. This yields that S2IC is complete with respect to DeV, which coupled with de Vries duality, yields that S2IC is the logic of compact Hausdorff spaces.

Our approach is closely related to that of Balbiani et al. [1], which along with [6] inspired the current paper. Balbiani et al. develop two-sorted logical calculi for region-based theories of space. Our calculus is

G. Bezhanishvili et al. / Annals of Pure and Applied Logic 170 (2019) 102714 3

simpler as we work with one-sorted propositional modal logic with one binary modality. In Section 7 we show how to translate the language of [1] into our language.

Two of the defining axioms of de Vries algebras are universal-existential statements or Π2-statements. They can be expressed in our language by use of non-standard rules, which we call Π2-rules. That S2ICis complete with respect to de Vries algebras shows that these Π2-rules are admissible in S2IC. This is again closely related to Balbiani et al. [1] who use similar non-standard rules in the context of region-based theories of space and prove their admissibility. In Section 6 we develop the theory of Π2-rules, show that they define inductive elementary subclasses of RSub, and that every derivation system axiomatized by Π2-rules is strongly sound and complete with respect to the subclass of RSub it defines. We also give a criterion of when a Π2-rule is admissible. We prove that being a zero-dimensional de Vries algebra is definable by a Π2-rule, and that this rule is admissible in S2IC. As a consequence, we obtain that S2IC is complete with respect to zero-dimensional de Vries algebra, and hence with respect to zero-dimensional compact Hausdorff spaces, also known as Stone spaces. On the other side of the spectrum from zero-dimensional spaces are connected spaces. We define the connected symmetric strict implication calculus CS2IC by adding one axiom to S2IC, and prove that CS2IC is complete with respect to connected de Vries algebras, and consequently with respect to connected compact Hausdorff spaces.

2. Subordinations, contact algebras, and de Vries algebras

In this section we recall the definitions of subordination, contact algebra, compingent algebra, and de Vries algebra, as well as the duality theory for these algebras. We also connect the duality theory for de Vries algebras to de Vries duality for compact Hausdorff spaces via Gleason spaces.

Definition 2.1. ([6])

(1) A subordination on a Boolean algebra B is a binary relation ≺ satisfying:(S1) 0 ≺ 0 and 1 ≺ 1;(S2) a ≺ b, c implies a ≺ b ∧ c;(S3) a, b ≺ c implies a ∨ b ≺ c;(S4) a ≤ b ≺ c ≤ d implies a ≺ d.

(2) We call (B, ≺) a subordination algebra, and let Sub be the class of all subordination algebras.

By Stone duality, Boolean algebras correspond to zero-dimensional compact Hausdorff spaces, known as Stone spaces. Given a Boolean algebra B, its dual Stone space is the space X of ultrafilters of B, the topology on which is given by the basis {β(a) | a ∈ B}, where β(a) = {x ∈ X | a ∈ x}. Then β is an isomorphism from B to the Boolean algebra Clop(X) of clopen subsets of X.

We say that a binary relation R on a Stone space X is closed if R is a closed subset of X × X in the product topology. Let StR be the class of pairs (X, R) where X is a Stone space and R is a closed relation on X. There is a one-to-one correspondence between Sub and StR, which extends to a categorical duality; see [6, Sec. 2.1]. This one-to-one correspondence can be obtained as follows. As usual, for a binary relation R on a set X and S ⊆ X, we write

R[S] := {x ∈ X | sRx for some s ∈ S}.

Let B be a Boolean algebra and X the Stone space of B. If R is a closed relation on X, then the binary relation ≺ defined by a ≺ b iff R[β(a)] ⊆ β(b) is a subordination on B. Conversely, let ≺ be a subordination on B. For S ⊆ B, write


�S := {a ∈ B | s ≺ a for some s ∈ S},

and define a binary relation R on X by xRy iff �x ⊆ y. Then R is a closed relation on X, and this correspondence is one-to-one.

We next consider the following additional properties of ≺:(S5) a ≺ b implies a ≤ b;(S6) a ≺ b implies ¬b ≺ ¬a;(S7) a ≺ b implies there is c ∈ B with a ≺ c ≺ b;(S8) a �= 0 implies there is b �= 0 with b ≺ a.

The next lemma gives a dual characterization of (S5)-(S7).

Lemma 2.2 ([13]). Let B be a Boolean algebra, X the Stone space of B, ≺ a subordination on B, and R the corresponding closed relation on X.

(1) (B, ≺) satisfies (S5) iff R is reflexive.(2) (B, ≺) satisfies (S6) iff R is symmetric.(3) (B, ≺) satisfies (S7) iff R is transitive.

Definition 2.3. Let (B, ≺) be a subordination algebra.

(1) We call (B, ≺) reflexive if (B, ≺) satisfies (S5), and let RSub be the class of reflexive subordination algebras.2

(2) ([27]) We call (B, ≺) a contact algebra if (B, ≺) satisfies (S5) and (S6), and let Con be the class of contact algebras.

We clearly have that Con ⊂ RSub ⊂ Sub, that reflexive subordination algebras dually correspond to the subclass of StR consisting of reflexive closed relations on Stone spaces, and that contact algebras dually correspond to the subclass of StR consisting of reflexive and symmetric closed relations on Stone spaces.

Definition 2.4.

(1) ([12]) We call a contact algebra (B, ≺) a compingent algebra if it satisfies (S7) and (S8).(2) ([4]) We call a compingent algebra (B, ≺) a de Vries algebra if B is a complete Boolean algebra.(3) Let Com be the class of compingent algebras and DeV the class of de Vries algebras.

We clearly have that DeV ⊂ Com ⊂ Con. Let (B, ≺) be a contact algebra and let (X, R) be its dual. As follows from Lemma 2.2, R is a reflexive and symmetric closed relation. Moreover, (B, ≺) satisfies (S7) iff Ris an equivalence relation. By [6, Lem. 6.3], (B, ≺) satisfies (S8) iff R is an irreducible equivalence relation, where we recall (see [6, Def. 6.1 and Rem. 6.2]) that R is irreducible provided R[U ] is a proper subset of Xfor each proper clopen subset U of X.

To characterize dually de Vries algebras we recall that a Boolean algebra B is complete iff its Stone space X is extremally disconnected, where a space is extremally disconnected provided the closure of each open set is clopen. Thus, compingent algebras dually correspond to pairs (X, R) where X is a Stone space and R is an irreducible equivalence relation, while de Vries algebras correspond to pairs (X, R) where Xis an extremally disconnected Stone space and R is an irreducible equivalence relation. Such pairs were

2 We point out that (B, ≺) being reflexive does not mean that ≺ is a reflexive relation, rather that the corresponding closed relation on the Stone space is reflexive (see Lemma 2.2(1)).


called Gleason spaces in [6, Def. 6.6] because of the close connection to Gleason covers of compact Hausdorff spaces.3

Let X be a compact Hausdorff space, and let (Y, π) be the Gleason cover of X. Define R on Y by xRy iff π(x) = π(y). Then (Y, R) is a Gleason space. Conversely, if (Y, R) is a Gleason space, then the quotient space X := Y/R is compact Hausdorff. This establishes a one-to-one correspondence between Gleason spaces and compact Hausdorff spaces, which extends to a categorical duality (see [6, Sec. 6] for details).

Since DeV dually corresponds to the class of Gleason spaces, it follows that DeV dually corresponds to the class KHaus of compact Hausdorff spaces, which is the object level of the celebrated de Vries duality [12]. The correspondence between DeV and KHaus can be obtained directly, as was done by de Vries.

For a compact Hausdorff space X, let RO(X) be the complete Boolean algebra of regular open subsets of X. Define ≺ on RO(X) by

U ≺ V iff Cl(U) ⊆ V.

Then (RO(X), ≺) is a de Vries algebra (that it validates (S7) and (S8) follows from the fact that every compact Hausdorff space is regular and normal; see, e.g., [14, Sec. 3.1]).

Conversely, suppose (B, ≺) is a compingent algebra. A round filter of (B, ≺) is a filter F of B satisfying

�F = F . An end of (B, ≺) is a maximal proper round filter. Let X be the set of ends of (B, ≺). For a ∈ B, let β(a) = {x ∈ X | a ∈ x}. Then {β(a) | a ∈ B} generates a compact Hausdorff topology on X. Moreover, if X is compact Hausdorff, then it is homeomorphic to the dual of (RO(X), ≺). If (B, ≺) is a compingent algebra and X is its dual, then (B, ≺) embeds into (RO(X), ≺), and (B, ≺) is isomorphic to (RO(X), ≺)iff (B, ≺) is a de Vries algebra. These correspondences extend to contravariant functors, which yield a dual equivalence of the categories KHaus and DeV. We refer to [12] for missing details and proofs.

3. The variety of strict implication algebras

As was pointed out in [6, Sec. 3], subordinations on B can be described by means of binary operations �: B ×B → B with values in {0, 1} satisfying(I1) 0 � a = a � 1 = 1;(I2) (a ∨ b) � c = (a � c) ∧ (b � c);(I3) a � (b ∧ c) = (a � b) ∧ (a � c).If ≺ is a subordination on B, then define �: B ×B → B by

a � b ={

1 if a ≺ b

0 otherwise.

It is easy to see that � has values in {0, 1} and satisfies (I1)–(I3). Conversely, given �, define ≺ by setting

a ≺ b iff a � b = 1.

It is easy to see that ≺ is a subordination on B, and that this correspondence is one-to-one.Moreover, the axioms (S5)–(S8) correspond, respectively, to the axioms:

(I4) a � b ≤ a → b;(I5) a � b = ¬b � ¬a;(I6) a � b = 1 implies ∃c : a � c = 1 and c � b = 1;

3 For details on Gleason covers we refer to [19] and [22, Sec. III.3]. They are not crucial for the content of this paper.


(I7) a �= 0 implies ∃b �= 0 : b � a = 1.Note that (I2)-(I3) correspond to (S2)-(S4) which explains why the numbering of the I-axioms is one off the numbering of the S-axioms. As we will see, adding (I4) to (I1)–(I3) is very useful in algebraic as well as logical calculations. Therefore, as our base variety, we will consider the variety generated by the algebras (B, �), where B is a Boolean algebra and � is a binary operation on B with values in {0, 1} satisfying (I1)–(I4). From now on, when we write (B, �) ∈ RSub, we mean that the corresponding (B, ≺) is reflexive (see Definition 2.3(1)).

Definition 3.1. We call (B, �) a strict implication algebra if (B, �) belongs to the variety generated by RSub. Let SIA be the variety of strict implication algebras.

Remark 3.2. While � is not a normal and additive operator on B, it gives rise to the normal and additive operator Δ(a, b) := ¬(a � ¬b). Then (B, Δ) is a BAO (Boolean algebra with operators), and � is definable from Δ by a � b = ¬Δ(a, ¬b). We prefer to work with � since it arises from subordinations more naturally.

Let (B, �) ∈ SIA. For a ∈ B, define

�a = 1 � a.

By (I1), �1 = 1 � 1 = 1; and by (I3), �(a ∧ b) = 1 � (a ∧ b) = (1 � a) ∧ (1 � b) = �a ∧�b. Thus, (B, �)is a (normal) modal algebra.

Suppose (B, �) ∈ RSub. If a = 1, then �a = 1. If a �= 1, then by (I4), �a = 1 � a ≤ 1 → a = a �= 1, so �a �= 1. But since (B, �) ∈ RSub, we have that � only takes values in {0, 1}. Therefore, � only takes values in {0, 1}. Thus, �a = 0, and so

�a ={

1 if a = 10 if a �= 1.

We let � be the dual of �, i.e., �a = ¬�¬a. Then

�a ={

0 if a = 01 if a �= 0,

and so � is the so-called unary discriminator term [21]. From this, and the fact that the class RSub is axiomatized by universal first-order formulas, the following observations are immediate [29, Sec. 8.2.].

Proposition 3.3.

(1) The variety SIA is a discriminator variety, and hence a semisimple variety.(2) The simple algebras in SIA are exactly the members of RSub.

We next turn to the axiomatization of SIA. First we observe that �a ≤ a by (I4). Let V be the variety of algebras (B, �) axiomatized by the equations defining Boolean algebras, (I1)–(I4), and the axioms:

(I8) �a ≤ ��a;(I9) ¬�a ≤ �¬�a;

(I10) a � b = �(a � b);(I11) �a ≤ ¬�a � 0.


We recall that a modal algebra (B, �) is an S5-algebra if its satisfies �a ≤ a, �a ≤ ��a, and ¬�a ≤ �¬�a. Thus, if (B, �) ∈ V, then (B, �) is an S5-algebra.

Theorem 3.4. SIA = V.

Proof. It is straightforward to see that (I8)–(I11) hold in each member of RSub. Since SIA is generated by RSub, it follows that SIA ⊆ V. For the reverse inclusion, we utilize [21, Thm. 3], by which a unary term �is a discriminator term in subdirectly irreducible members of a variety V iff V satisfies four equations that in our setting amount to:

• �a ≤ ��a;• ¬�a ≤ �¬�a;• �a ≤ ¬�a � 0;• ¬�a ≤ �a � 0.

Clearly each (B, �) ∈ V satisfies the first three. To see that it also satisfies the fourth, observe that by (I4) and (I9) we have ¬�a = �¬�a. This together with (I11) yields

¬�a = �¬�a ≤ ¬�¬�a � 0 = ¬¬�a � 0 = �a � 0.

Thus, [21, Thm. 3] applies, by which � is a unary discriminator in all subdirectly irreducible members of V. So if (B, �) is a subdirectly irreducible member of V, then � only takes the values 0 and 1. Since (I10) holds in (B, �), we have that � also only takes the values 0 and 1. Therefore, as (I1)–(I4) hold in (B, �), it follows from the definition of RSub that (B, �) ∈ RSub. Thus, each subdirectly irreducible member of Vbelongs to SIA. Since V is generated by its subdirectly irreducible algebras, we conclude that V ⊆ SIA. �

We next give an alternate axiomatization of SIA, which will be useful in Section 4. Let W be the variety axiomatized by (I1)–(I4) and the axioms:

(I12) �(a → b) ∧ (b � c) ≤ a � c;(I13) (a � b) ∧ �(b → c) ≤ a � c;(I14) a � b ≤ c � (a � b);(I15) ¬(a � b) ≤ c � ¬(a � b).

To prove that SIA = W, we require the following lemma.

Lemma 3.5. (I2) and (I3) imply a ≤ b ⇒ (b � c ≤ a � c and c � a ≤ c � b).

Proof. By (I2), b � c = (a ∨ b) � c = (a � c) ∧ (b � c). Therefore, b � c ≤ a � c. Also, by (I3), c � a = c � (a ∧ b) = (c � a) ∧ (c � b). Thus, c � a ≤ c � b. �Theorem 3.6. SIA = W.

Proof. First we show that SIA ⊆ W. For this it is sufficient to see that (I12)–(I15) hold in each strict implication algebra (B, �). To see that (I12) holds, by (I11),

�(a → b) ∧ (b � c) ≤ (¬�(a → b) � 0) ∧ (b � c).

Since 0 ≤ c, by Lemma 3.5 and (I2),


(¬�(a → b) � 0) ∧ (b � c) ≤ (¬�(a → b) � c) ∧ (b � c)

= (¬�(a → b) ∨ b) � c

= (�(a → b) → b) � c.

Because �(a → b) ≤ a → b, we have (a → b) → b ≤ �(a → b) → b. Therefore, by Lemma 3.5,

(�(a → b) → b) � c ≤ ((a → b) → b) � c.

But (a → b) → b = ¬(¬a ∨ b) ∨ b = a ∨ b, so applying Lemma 3.5 again yields

((a → b) → b) � c = (a ∨ b) � c ≤ a � c.

Thus, (I12) holds in (B, �).To see that (I13) holds, by Lemma 3.5 and (I3),

(a � b) ∧ �(b → c) = (a � b) ∧ (1 � (b → c))

≤ (a � b) ∧ (a � (b → c))

= a � (b ∧ (b → c)).

Since b ∧ (b → c) ≤ c, applying Lemma 3.5 again yields

a � (b ∧ (b → c)) ≤ a � c.

Thus, (I13) holds in (B, �).To see that (I14) holds, by (I10) and Lemma 3.5,

a � b = �(a � b) = 1 � (a � b) ≤ c � (a � b).

Thus, (I14) holds in (B, �).To see that (I15) holds, since (B, �) is an S5-algebra, by (I10) and Lemma 3.5,

¬(a � b) = ¬�(a � b) = �¬�(a � b)

= �¬(a � b) = 1 � ¬(a � b)

≤ c � ¬(a � b).

Thus, (I15) holds in (B, �). Consequently, SIA ⊆ W.It is left to show that W ⊆ SIA. For this it is sufficient to see that (I8)–(I11) hold in each (B, �) ∈ W.

It follows from (I14) that a � b ≤ �(a � b), and it follows from (I4) that �(a � b) ≤ a � b. Thus, (I10) holds in (B, �).

That (I8) holds in (B, �) is immediate from (I10):

�a = 1 � a = �(1 � a) = ��a.

To see that (I11) holds, substituting in (I12) ¬�a for a and 0 for both b and c yields �(¬�a → 0) ∧ (0 �0) ≤ ¬�a � 0. Now, using (I1) and (I8), we have:

�(¬�a → 0) ∧ (0 � 0) = �(¬¬�a) ∧ 1 = ��a = �a.


Thus, �a ≤ ¬�a � 0, and so (I11) holds in (B, �).Finally, it follows from (I15) that ¬(a � b) ≤ �¬(a � b). Thus,

¬�a = ¬(1 � a) ≤ �¬(1 � a) = �¬�a,

and hence (I9) holds in (B, �). Consequently, W ⊆ SIA. �We next show that our base variety SIA is locally finite, and consider subvarieties and inductive subclasses

of SIA.

Proposition 3.7. The variety SIA is locally finite.

Proof. Let (B, �) ∈ RSub be n-generated, with generators a1, . . . , an ∈ B. For each a ∈ B, there is a term t(x1, . . . , xn) such that a = t(a1, . . . , an). Since (B, �) ∈ RSub, for each b, c ∈ B, we have b � c ∈ {0, 1}. Therefore, by replacing each subterm of t(x1, . . . , xn) of the form x � y with either 0 or 1, we obtain a Boolean term t′(x1, . . . , xn) such that a = t′(a1, . . . , an). Thus, B is n-generated as a Boolean algebra, and hence has at most 22n elements. Since RSub is the class of simple algebras in SIA, which is a semisimple variety (see Proposition 3.3), there is a uniform bound m(n) = 22n on all n-generated subdirectly irreducible members of SIA. Consequently, by [3, Thm. 3.7(4)], SIA is locally finite. �

As an immediate consequence we obtain:

Corollary 3.8. Every subvariety of SIA is generated by its finite members.

While SIA has many subvarieties, we will be interested in the subvariety obtained by postulating the identity (I5). Our interest is motivated by the fact that this variety is exactly the subvariety of SIA generated by the class Con of contact algebras. We further restrict Con to the class Com of compingent algebras, by postulating (I6) and (I7). But unlike (I5), neither (I6) nor (I7) is an identity. However, both (I6) and (I7) are Π2-statements (i.e., statements of the form ∀x∃yΦ(x, y), where x, y are tuples of variables and Φ(x, y) is a quantifier-free formula). By the Chang-Łoś-Suszko Theorem (see, e.g., [11, Thm. 3.2.3]), the elementary classes corresponding to Π2-statements are inductive classes, where we recall that a class is inductive provided it is closed under unions of chains (equivalently, closed under directed limits). While we will be mainly interested in the inductive class Com, in Section 6 we will show that all elementary inductive subclasses of RSub can be axiomatized by non-standard rules.

We conclude this section by observing that, unlike subvarieties of SIA, not every inductive subclass of SIA is determined by its finite algebras. For example, the inductive elementary class Com is not determined by its finite algebras. To see this, let Dis be the subclass of Com consisting of those algebras in Com that validate the equation a � a = 1. Then Dis is an inductive elementary subclass of Com. To see that Dis is a proper subclass of Com, let X = [0, 1]. Then the de Vries algebra (RO(X), ≺) falsifies a � a = 1. Indeed, if we put a = [0, 12 ), then the closure of a is [0, 12 ] � a. So a ⊀ a, and hence a � a �= 1. On the other hand, we show that every finite algebra in Com validates the equation a � a = 1. Since every finite compingent algebra is a finite de Vries algebra, by de Vries duality, every finite compingent algebra (B, ≺) is isomorphic to the powerset of a finite discrete space X. Because every subset of a discrete space is clopen, we have a ≺ a, so a � a = 1 for each a ∈ B. Therefore, Dis and Com have the same finite algebras, yet Dis is a proper subclass of Com. Thus, Com is not determined by its finite algebras.

4. The strict implication calculus

We next present a sound and complete deductive system for SIA. We will work with the language of clas-sical propositional logic, with a countably infinite supply of propositional letters and primitive connectives


∧, ¬, which we will enrich with one binary connective � of strict implication. Then �, ⊥, ∨, →, ↔ are usual abbreviations, and �ϕ abbreviates � � ϕ.

A valuation on (B, �) is an assignment of elements of B to propositional letters of our language L, which extends to all formulas of L in the usual way. We say that a valuation v on (B, �) satisfies a formula ϕif v(ϕ) = 1. In such a case we write (B, �, v) |= ϕ. If all valuations on (B, �) satisfy ϕ, then we say that (B, �) validates ϕ, and write (B, �) |= ϕ. For a set of formulas Γ, we write (B, �) |= Γ if (B, �) |= ϕ for every ϕ ∈ Γ.

Suppose U ⊆ SIA, ϕ is a formula, and Γ is a set of formulas. We say that ϕ is a semantic consequence of Γ over U , and write Γ |=U ϕ, provided for each (B, �) ∈ U and each valuation v on (B, �), if v(γ) = 1 for each γ ∈ Γ, then v(ϕ) = 1.

Consider the following axiom schemes:(A1) (⊥ � ϕ) ∧ (ϕ � �),(A2) [(ϕ ∨ ψ) � χ] ↔ [(ϕ � χ) ∧ (ψ � χ)],(A3) [ϕ � (ψ ∧ χ)] ↔ [(ϕ � ψ) ∧ (ϕ � χ)],(A4) (ϕ � ψ) → (ϕ → ψ),(A5) (ϕ � ψ) ↔ (¬ψ � ¬ϕ),(A8) �ϕ → ��ϕ,(A9) ¬�ϕ → �¬�ϕ,(A10) (ϕ � ψ) ↔ �(ϕ � ψ),(A11) �ϕ → (¬�ϕ � ⊥),(A12) [�(ϕ → ψ) ∧ (ψ � χ)] → (ϕ � χ),(A13) [(ϕ � ψ) ∧ �(ψ → χ)] → (ϕ � χ),(A14) (ϕ � ψ) → [χ � (ϕ � ψ)],(A15) ¬(ϕ � ψ) → [χ � ¬(ϕ � ψ)].

Clearly (A1)–(A5) correspond to (I1)–(I5) and (A8)–(A15) to (I8)–(I15).

Definition 4.1. The strict implication calculus SIC is the derivation system containing:

• all the theorems of the classical propositional calculus CPC,• the axiom schemes (A1)–(A5) and (A8)–(A11),

and closed under the inference rules:

(MP) ϕ ϕ → ψ

ψ

(N) ϕ

�ϕ

The definition of derivability in SIC is standard:

Definition 4.2.

(1) A proof of a formula ϕ from a set of formulas Γ is a finite sequence ψ1, . . . , ψn such that ψn = ϕ and each ψi is in Γ or is an instance of an axiom of SIC or is obtained from ψj, ψk for some j, k < i by applying (MP), or is obtained from ψj for some j < i by applying (N). Elements of Γ are referred to as assumptions.

(2) If there is a proof of ϕ from Γ, then we say that ϕ is derivable in SIC from Γ and write Γ �SIC ϕ.(3) If Γ = ∅, then we say that ϕ is derivable in SIC and write �SIC ϕ.


Remark 4.3. Since �(ϕ ∧ ψ) ↔ (�ϕ ∧ �ψ) is an instance of (A3) and �ϕ → ϕ is an instance of (A4), we see that all the theorems of the modal system S5 are derivable in SIC. In particular, the (K) axiom �(ϕ → ψ) → (�ϕ → �ψ) is derivable in SIC.

The deduction theorem for SIC is proved as for S5:

Theorem 4.4. For any set of formulas Γ and for any formulas ϕ, ψ, we have:

Γ ∪ {ϕ} �SIC ψ ⇔ Γ �SIC �ϕ → ψ.

Proof. (⇐) This is the easy direction since a proof ψ1, . . . , ψn of �ϕ → ψ from Γ can easily be extended to a proof of ψ from Γ ∪ {ϕ} as follows:

n. �ϕ → ψ

n + 1. ϕ (assumption)n + 2. �ϕ (by (N) from n + 1)n + 3. ψ (by (MP) from n and n + 2).

(⇒) Suppose there is a proof ψ1, . . . , ψn of ψ from Γ ∪ {ϕ}. We show by induction on i = 1, . . . , n that we can obtain a proof of �ϕ → ψi from Γ. If ψi = ϕ, then Γ �SIC �ϕ → ψi since �SIC �ϕ → ϕ. If ψi ∈ Γor ψi is an instance of an axiom of SIC, then since �SIC ψi → (�ϕ → ψi), by applying (MP) we obtain Γ �SIC �ϕ → ψi. If ψi is obtained by applying (MP) to ψj and ψk = ψj → ψi with j, k < i, then by the inductive hypothesis, Γ �SIC �ϕ → ψj , �ϕ → (ψj → ψi). But then Γ �SIC �ϕ → ψj , ψj → (�ϕ → ψi), which yields Γ �SIC �ϕ → (�ϕ → ψi), so Γ �SIC �ϕ → ψi. Finally, if ψi is obtained by applying (N) to ψj

with j < i, then by the inductive hypothesis, Γ �SIC �ϕ → ψj . Applying (N) yields Γ �SIC �(�ϕ → ψi). Therefore, by applying the (K) axiom (see Remark 4.3) and (MP), we obtain Γ �SIC ��ϕ → �ψj . Thus, since �SIC �ϕ → ��ϕ, we have Γ �SIC �ϕ → �ψj , and so Γ �SIC �ϕ → ψi, concluding the proof. �

Since each axiom of SIC has an equational counterpart in the axiomatization of SIA, the standard Lin-denbaum construction (see, e.g., [23]) yields the following.

Proposition 4.5. SIC is strongly sound and complete with respect to SIA; that is, for a set of formulas Γ and a formula ϕ,

Γ �SIC ϕ iff Γ |=SIA ϕ.

Remark 4.6. As follows from Theorem 3.6, SIA can be axiomatized by replacing (I8)–(I11) with (I12)–(I15). Thus, by Proposition 4.5, SIC can be axiomatized by replacing (A8)–(A11) with (A12)–(A15).

We next show that SIC is in fact strongly sound and complete with respect to RSub. For this we first characterize congruences of strict implication algebras. It is well known that congruences of Boolean algebras correspond to filters, and this correspondence is obtained as follows. If θ is a congruence on a Boolean algebra B, then Fθ = {a ∈ B | aθ1} is a filter of B. If F is a filter of B, then θF defined by aθF b iff a ↔ b ∈ F is a congruence of B. Moreover, θFθ

= θ and FθF = F .

Proposition 4.7. For (B, �) ∈ SIA, there is a one-to-one correspondence between(1) congruences of (B, �);(2) congruences θ of B such that aθb implies (a � c)θ(b � c) and (c � a)θ(c � b);(3) filters F of B such that a ∈ F implies �a ∈ F ;


(4) filters F of B such that a → b ∈ F implies (b � c) → (a � c), (c � a) → (c � b) ∈ F ;(5) filters F of B such that a → b, b � c, c → d ∈ F imply a � d ∈ F .

Proof. (1)⇒(2) This is obvious.(2)⇒(1) Suppose aθb and cθd. By (2), (a � c)θ(b � c) and (b � c)θ(b � d). Therefore, (a � c)θ(b � d).

Thus, θ is a congruence of (B, �).(3)⇒(4) Suppose F satisfies (3) and a → b ∈ F . Then �(a → b) ∈ F . By (I12) and (I13), for any

c ∈ B, we have �(a → b) ≤ (b � c) → (a � c) and �(a → b) ≤ (c � a) → (c � b). Therefore, (b � c) → (a � c), (c � a) → (c � b) ∈ F , and so F satisfies (4).

(4)⇒(5) Suppose F satisfies (4) and a → b, b � c, c → d ∈ F . From a → b ∈ F it follows that (b � c) → (a � c) ∈ F . Therefore, since b � c ∈ F , we have a � c ∈ F . Also, from c → d ∈ F it follows that (a � c) → (a � d) ∈ F . This together with a � c ∈ F yields a � d ∈ F . Thus, F satisfies (5).

(5)⇒(3) Suppose F satisfies (5) and a ∈ F . Since 1 → 1 = 1 � 1 = 1 and 1 → a = a, we have 1 → 1, 1 � 1, 1 → a ∈ F . Therefore, by (5), �a = 1 � a ∈ F . Thus, F satisfies (3).

(2)⇒(3) Suppose θ is a congruence of B and a ∈ Fθ. Then aθ1. Therefore, (1 � a)θ(1 � 1). Thus, �aθ1, and so �a ∈ Fθ.

(4)⇒(2) Suppose F satisfies (4), aθF b, and c ∈ B. Then a → b ∈ F and b → a ∈ F . Therefore, by (4), (b � c) → (a � c), (c � a) → (c � b) ∈ F and (a � c) → (b � c), (c � b) → (c � a) ∈ F . Thus, (a � c) ↔ (b � c), (c � a) ↔ (c � b) ∈ F . Consequently, (a � c)θF (b � c) and (c � a)θF (c � b), and hence θF satisfies (2). �Definition 4.8. Let (B, �) be a strict implication algebra. We call a filter F of B a �-filter provided Fsatisfies Proposition 4.7(3); that is, a ∈ F implies �a ∈ F .

By Proposition 4.7, congruences of strict implication algebras correspond to their �-filters. This is a generalization of a similar characterization of congruences of modal algebras (see, e.g., [10, Sec. 7.7]). For a strict implication algebra (B, �) and a ∈ B, we use the usual abbreviation

↑a := {b ∈ B | a ≤ b}.

Lemma 4.9. Let (B, �) be a strict implication algebra, a ∈ B, and F a �-filter. Then the filter generated by F ∪ {�a} is a �-filter. In particular, we have that ↑�a and ↑¬�a are �-filters.

Proof. Let F ′ be the filter generated by F ∪{�a}, and let b ∈ F ′. Then there is c ∈ F such that c ∧�a ≤ b. As � is an S5-operator, we have �(c ∧ �a) ≤ �b. Since F is a �-filter, �c ∈ F . Therefore, again using the fact that � is an S5-operator, we obtain �(c ∧ �a) = �c ∧ ��a = �c ∧ �a ∈ F ′. Thus, �b ∈ F ′, which shows that F ′ is a �-filter.

In particular, as {1} is a �-filter, it follows that ↑�a is a �-filter, and by (I9) the same holds for ↑¬�a. �For a strict implication algebra (B, �) and a �-filter F , let (B/F, �F ) be the quotient algebra. For

a ∈ B, we let [a] be the corresponding element of B/F .

Lemma 4.10. Let (B, �) ∈ SIA.(1) For a proper �-filter F in (B, �), the following are equivalent:

(a) F is a maximal proper �-filter.(b) For each a ∈ B, we have �a ∈ F or ¬�a ∈ F .(c) (B/F, �F ) ∈ RSub.

(2) If F is a �-filter and a /∈ F , then there is a maximal �-filter M such that F ⊆ M and a /∈ M .


Proof. (1) (a)⇒(b) Suppose �a /∈ F . Let G be the filter generated by F and �a. By Lemma 4.9, G is a �-filter. Since F is a maximal �-filter, G is improper. Therefore, 0 = �a ∧ b for some b ∈ F . Thus, b ≤ ¬�a, and so ¬�a ∈ F .

(b)⇒(c) Let a ∈ B. Then �a ∈ F or ¬�a ∈ F . If �a ∈ F , then �F [a] = [�a] = 1F , where �F [a] =1F �F [a]. On the other hand, if �a /∈ F , then ¬�a ∈ F , so ¬F�F [a] = [¬�a] = 1F , and hence �F [a] = 0F . This implies that {1F } and B/F are the only two �F -filters in (B/F, �F ). Thus, (B/F, �F ) is a simple algebra, and hence (B/F, �F ) ∈ RSub by Proposition 3.3(2).

(c)⇒(a) Suppose G is a �-filter properly containing F . Then there is a ∈ G \ F . Since G is a �-filter and �a ≤ a, we see that �a ∈ G \ F . Therefore, [�a] �= 1F . Since (B/F,�F ) ∈ RSub, we conclude that [�a] = 0F . Thus, [¬�a] = 1F , yielding that ¬�a ∈ F ⊆ G. Consequently, G is an improper �-filter, and hence F is a maximal �-filter.

(2) Since a /∈ F , by Zorn’s lemma there is a �-filter M such that F ⊆ M , a /∈ M , and M is maximal with this property. If M is not a maximal �-filter, then by (1), there is b ∈ B such that �b, ¬�b /∈ M . Let G be the filter generated by M and �b and H the filter generated by M and ¬�b. By Lemma 4.9, both G and H are �-filters that properly extend M . Therefore, a ∈ G, H, so there exist c, d ∈ M such that a ≥ �b ∧ c

and a ≥ ¬�b ∧ d. Thus, a ≥ (�b ∧ c) ∨ (¬�b ∧ d) = (�b ∨ ¬�b) ∧ (�b ∨ d) ∧ (c ∨ ¬�b) ∧ (c ∨ d) ∈ M . The obtained contradiction proves that M is a maximal �-filter. �Theorem 4.11. For a set of formulas Γ and a formula ϕ, we have:

Γ �SIC ϕ ⇔ Γ |=SIA ϕ ⇔ Γ |=RSub ϕ.

Proof. We already observed in Proposition 4.5 that Γ �SIC ϕ ⇔ Γ |=SIA ϕ. This together with RSub ⊆ SIAyields that Γ �SIC ϕ implies Γ |=RSub ϕ. Conversely, if Γ �SIC ϕ, then in the Lindenbaum algebra (B, �) of SIC, the �-filter generated by {[ψ] | ψ ∈ Γ} does not contain [ϕ]. By Lemma 4.10(2), there is a maximal �-filter F such that {[ψ] | ψ ∈ Γ} ⊆ F and [ϕ] /∈ F . But then (B/F, �F ) satisfies Γ and refutes ϕ. By Lemma 4.10(1), (B/F, �F ) ∈ RSub. Thus, Γ �|=RSub ϕ. �5. The symmetric strict implication calculus and its topological completeness

In this section we define the symmetric strict implication calculus S2IC obtained by adding (A5) to SIC, and the corresponding variety S2IA of symmetric strict implication algebras. We prove that S2IC is strongly sound and complete with respect to Con, as well as with respect to Com and DeV. The last completeness together with de Vries duality allows us to introduce topological models for S2IC based on compact Hausdorff spaces, and prove that S2IC is strongly sound and complete with respect to the class of compact Hausdorff spaces.

Definition 5.1.

(1) We call a strict implication algebra (B, �) symmetric if it satisfies (I5). Let S2IA be the variety of symmetric strict implication algebras.

(2) The symmetric strict implication calculus S2IC is obtained from the strict implication calculus SIC by postulating (A5).

Since (A5) corresponds to (I5), it follows from Proposition 4.5 that S2IC is strongly sound and complete with respect to S2IA. Moreover, since for (B, �) ∈ RSub we have (B, �) ∈ Con iff (B, �) satisfies (I5), the following is an immediate consequence of Theorem 4.11.


Theorem 5.2. For a set of formulas Γ and a formula ϕ, we have:

Γ �S2IC ϕ ⇔ Γ |=S2IA ϕ ⇔ Γ |=Con ϕ.

We next show that each contact algebra can be embedded into a compingent algebra. For this we uti-lize the representation of subordination algebras discussed in Section 2, as well as the following result (cf. Lemma 2.2).

Lemma 5.3 ([13]). Let R be a binary relation on a set X. Define ≺R on P(X) by U ≺R V iff R[U ] ⊆ V .(1) ≺R is a subordination on P(X).(2) R is reflexive iff (P(X), ≺R) satisfies (S5).(3) R is symmetric iff (P(X), ≺R) satisfies (S6).(4) R is transitive iff (P(X), ≺R) satisfies (S7).

We use Lemma 5.3 to show an analogue of [1, Lem. 2.5] in our setting. Let (B, ≺) and (C, ≺) be in RSub. We say that (B, ≺) is embedded into (C, ≺) if there is a Boolean embedding h : B → C such that a ≺ b iff h(a) ≺ h(b) for each a, b ∈ B.

Lemma 5.4.

(1) Every (B, ≺) ∈ RSub can be embedded into some (C, ≺) ∈ RSub satisfying (S7).(2) Every (B, ≺) ∈ Con can be embedded into some (C, ≺) ∈ Con satisfying (S7).

Proof. (1) Suppose that (X, R) is the dual of (B, ≺). By Lemma 2.2(1), R is reflexive. Let Y = {{x, y} ⊆X | xRy} and let

X ′ = {(x, α) ∈ X × Y | x ∈ α}.

Define R′ on X ′ by

(x, α)R′(y, β) ⇔ xRy and α = β.

We show that R′ is reflexive and transitive. That R′ is reflexive follows from the reflexivity of R. To see that R′ is transitive, let (x, α)R′(y, β)R′(z, γ). Then xRyRz and α = β = γ. Therefore, either x = y, y = z, or z = x. Since R is reflexive, we see that in each of these cases we have xRz. Thus, (x, α)R′(z, γ), and so R′ is transitive.

Define f : X ′ → X by f(x, α) = x. Clearly f is onto. Therefore, f−1 : Clop(X) → P(X ′) is a Boolean embedding.

Claim. For U, V ∈ Clop(X), we have U ≺R V iff f−1(U) ≺R′ f−1(V ).

Proof of claim. It follows from the definition of R′ that (x, α)R′(y, β) implies f(x, α)Rf(y, β). So U ≺R V

implies f−1(U) ≺R′ f−1(V ). For the converse, suppose U ⊀R V . Then R[U ] � V . Therefore, there are x ∈ U and y /∈ V such that xRy. Let α = {x, y}. Then (x, α)R′(y, α), (x, α) ∈ f−1(U), and (y, α) /∈ f−1(V ). Thus, R′[f−1(U)] � f−1(V ), and hence f−1(U) ⊀R′ f−1(V ). �

Let (C, ≺) = (P(X ′), ≺R′). By Lemma 5.3, (C, ≺) satisfies (S1)–(S5) and (S7), and by the Claim, f−1 is an embedding of (B, ≺) into (C, ≺).

(2) If (B, ≺) ∈ Con, then by Lemma 2.2(2), R is also symmetric. Therefore, so is R′, and hence R′ is an equivalence relation. Thus, by Lemma 5.3, (C, ≺) satisfies (S1)–(S7), concluding the proof. �


Lemma 5.5. Suppose (B, ≺) ∈ RSub. Let B′ = B ×B and define ≺′ on B′ by

(a, b) ≺′ (c, d) ⇔ a ≺ c and b ≤ d.

Then (B′, ≺′) ∈ RSub. Moreover, if (B, ≺) ∈ Con, then (B′, ≺′) ∈ Con.

Proof. Since (B, ≺) ∈ RSub, it satisfies (S1)–(S5). We show that (B′, ≺′) also satisfies (S1)–(S5).(S1) Since 0 ≺ 0 and 1 ≺ 1, it is obvious that (0, 0) ≺′ (0, 0) and (1, 1) ≺′ (1, 1).(S2) Suppose (a, b) ≺ (c, d), (c′, d′). Then a ≺ c, c′ and b ≤ d, d′. Therefore, a ≺ c ∧ c′ and b ≤ d ∧ d′.

Thus, (a, b) ≺′ (c, c′) ∧ (d, d′).(S3) Suppose (a, b), (a′, b′) ≺′ (c, d). Then a, a′ ≺ c and b, b′ ≤ d. Therefore, a ∨ a′ ≺ c and b ∨ b′ ≤ d.

Thus, (a, b) ∨ (a′, b′) ≺′ (c, d).(S4) Suppose (a, b) ≤ (a′, b′) ≺ (c′, d′) ≤ (c, d). Then a ≤ a′ ≺ c′ ≤ c and b ≤ b′ ≤ d′ ≤ d. Thus, a ≺ c

and b ≤ d, and so (a, b) ≺′ (c, d).(S5) Suppose (a, b) ≺′ (c, d). Then a ≺ c and b ≤ d. From a ≺ c it follows that a ≤ c. Thus, (a, b) ≤ (c, d).Now suppose that in addition (B, ≺) ∈ Con. Then (B, ≺) satisfies (S6). We show that (B′, ≺′) also

satisfies (S6).(S6) Suppose (a, b) ≺′ (c, d). Then a ≺ c and b ≤ d. Therefore, ¬c ≺ ¬a and ¬d ≤ ¬b. Thus, ¬(c, d) ≺′

¬(a, b).Consequently, if (B, ≺) ∈ Con, then (B′, ≺′) ∈ Con. �

Lemma 5.6.

(1) Every (B, ≺) ∈ RSub can be embedded into (C, ≺) ∈ RSub satisfying (S8).(2) In addition, if (B, ≺) satisfies either (S6) or (S7), then so does (C, ≺).

Proof. (1) Starting from (B, ≺), we inductively build a chain

(B,≺) ↪→ (B1,≺) ↪→ (B2,≺) ↪→ (B3,≺) ↪→ · · ·

in RSub such that the union (C, ≺) :=⋃

n∈ω(Bn, ≺) satisfies (S8).If (Bn, ≺) is already defined, define (Bn+1, ≺) := (Bn, ≺) × (Bn, ≤). By Lemma 5.5, (Bn+1, ≺) ∈ RSub.

Moreover, a �→ (a, a) is an embedding of (Bn, ≺) into (Bn+1, ≺). We prove that (C, ≺) satisfies (S8).Let 0 �= a ∈ C. Then there is n such that a ∈ Bn. Therefore, (a, a) ∈ Bn+1. Let b := (0, a) ∈ Bn+1. We

have b �= 0 and b ≺ (a, a). Thus, (C, ≺) satisfies (S8).(2) If (B, ≺) satisfies (S6), then each (Bn, ≺) satisfies (S6) by Lemma 5.5. Therefore, so does (C, ≺).Finally, suppose that in addition (B, ≺) satisfies (S7). We show that if (Bn, ≺) satisfies (S7), then so

does (Bn+1, ≺). Let (a1, a2) ≺ (b1, b2) in (Bn+1, ≺). Then a1 ≺ b1 and a2 ≤ b2 in Bn. By (S7), there exists c ∈ Bn such that a1 ≺ c and c ≺ b1. So, for (c, a2) ∈ Bn+1, we have (a1, a2) ≺ (c, a2) ≺ (b1, b2). Therefore, (Bn+1, ≺) satisfies (S7). Thus, by induction, each (Bn, ≺) satisfies (S7). By the Chang-Łoś-Suszko theorem (see, e.g., [11, Thm. 3.2.3]), Π2-sentences are preserved by direct limits. Since (S7) is a Π2-sentence, the direct limit (C, ≺) of the chain also satisfies (S7). �

Let (B, �) and (C, �) be the strict implication algebras corresponding to (B, ≺) and (C, ≺), respectively. It is straightforward to check that h : B → C is an embedding of (B, ≺) into (C, ≺) iff h is an isomorphism from (B, �) to a subalgebra of (C, �). For a class K of strict implication algebras, let IS(K) be the class of isomorphic copies of subalgebras of algebras in K.

Theorem 5.7. IS(Com) = Con.


Proof. Obviously, Com ⊆ Con and as Con is a universal class, we have that IS(Com) ⊆ Con. Conversely, suppose (B, �) ∈ Con. Then by Lemmas 5.4(2) and 5.6 it is isomorphic to a subalgebra of (C, �) ∈ Com. Therefore, Con ⊆ IS(Com). �Theorem 5.8. S2IC is strongly sound and complete with respect to Com i.e., for a set of formulas Γ and a formula ϕ, we have:

Γ �S2IC ϕ ⇔ Γ |=Com ϕ.

Proof. The left to right direction follows from Theorem 5.2 and the fact that Com ⊆ Con. Now suppose Γ �S2IC. Applying Theorem 5.2 again yields a contact algebra (B, �) and a valuation v on B such that v(γ) = 1B for each γ ∈ Γ and v(ϕ) �= 1B . By Theorem 5.7, there is (C, �) ∈ Com such that (B, �) is isomorphic to a subalgebra of (C, �). We may view v as a valuation on C, so v(γ) = 1C for each γ ∈ Γ and v(ϕ) �= 1C . Thus, Γ �|=Com ϕ. �

We recall that the de Vries algebra of a compact Hausdorff space X is the pair (RO(X), ≺), where RO(X) is the complete Boolean algebra of regular open subsets of X and U ≺ V iff Cl(U) ⊆ V . By de Vries duality [12], every de Vries algebra is isomorphic to the de Vries algebra of some compact Hausdorff space. This allows us to define topological semantics for our language.

Definition 5.9. A compact Hausdorff model is a pair (X, v), where X is a compact Hausdorff space and v is a valuation assigning a regular open set to each propositional letter.

If � is the strict implication corresponding to ≺, then the formulas of our language are interpreted in (RO(X), �) ∈ DeV.

Theorem 5.10.

(1) The system S2IC is strongly sound and complete with respect to DeV.(2) The system S2IC is strongly sound and complete with respect to compact Hausdorff models.

Proof. (1) Since DeV ⊆ Com, by Theorem 5.8, we have Γ �S2IC ϕ implies Γ |=DeV ϕ. Conversely, suppose Γ �S2IC ϕ. Applying Theorem 5.8 again yields a compingent algebra (B, �) and a valuation v on B such that v(γ) = 1B for each γ ∈ Γ and v(ϕ) �= 1B . By de Vries duality, there is a compact Hausdorff space Xsuch that (B, �) embeds into (RO(X), �). We may view v as a valuation on RO(X), so v(γ) = X for each γ ∈ Γ and v(ϕ) �= X. Since (RO(X), �) ∈ DeV, we conclude that Γ �|=DeV ϕ.

(2) This follows from (1) and de Vries duality. �Remark 5.11. Let (B, ≺) ∈ Com and let β : B → RO(X) be the embedding. By [12, Thm. I.3.9], for U, V ∈ RO(X) with U ≺ V , there are a, b ∈ B with a ≺ b, U ⊆ β(a), and β(b) ⊆ V . From this it follows that RO(X) is isomorphic to the MacNeille completion of B. Thus, it is possible to prove Theorem 5.10(1) without using the de Vries representation of compingent algebras. Namely, for (B, ≺) ∈ Com, let B be the MacNeille completion of B. By identifying B with its image, we may view B as a subalgebra of B, and define � on B by setting

x � y iff there exist a, b ∈ B such that x ≤ a ≺ b ≤ y.

A direct verification shows that (B, �) ∈ DeV, which yields a point-free proof of Theorem 5.10(1); see [5, Lem. 6.5] for details.


6. Π2-rules, admissibility, and further completeness results

6.1. Π2-rules

As we saw in the previous section, S2IC is strongly sound and complete with respect to DeV, and hence S2IC is the strict implication logic of compact Hausdorff models. Note that neither (I6) nor (I7) is expressible in our logic as S2IC is also strongly sound and complete with respect to Con. This generates an interesting question of what logical formalism to use when reasining about compact Hausdorff models (see Remark 6.23for more discussion). In this section we show that we can express (I6) and (I7) in our propositional language by means of Π2-rules. For this we first rewrite (I6) and (I7) in the following form.

(Π6) ∀x1, x2, y(x1 � x2 � y → ∃z : (x1 � z) ∧ (z � x2) � y

);

(Π7) ∀x, y(x � y → ∃z : z ∧ (z � x) � y

)

Lemma 6.1. Let (B, �) ∈ RSub.(1) (B, �) |= (I6) iff (B, �) |= (Π6).(2) (B, �) |= (I7) iff (B, �) |= (Π7).

Proof. (1) (⇒) Suppose (B, �) |= (I6). Let a, b, d ∈ B be such that a � b � d. Then d �= 1 and a � b �= 0, so a � b = 1 since (B, �) ∈ RSub. By (I6), there is c ∈ B such that a � c = c � b = 1. Therefore, 1 = (a � c) ∧ (c � b) � d. Thus, (B, �) |= (Π6).

(⇐) Suppose (B, �) |= (Π6). Let a, b ∈ B be such that a � b = 1. Then a � b � 0. By (Π6), there is c ∈ B such that (a � c) ∧ (c � b) � 0. Therefore, since (B, �) ∈ RSub, we have a � c = c � b = 1. Thus, (B, �) |= (I6).

(2) (⇒) Suppose (B, �) |= (I7). Let a, c ∈ B be such that a � c. Then a ∧ ¬c �= 0. By (I7), there is b �= 0 such that b � (a ∧ ¬c) = 1. By (I3), b � (a ∧ ¬c) = (b � a) ∧ (b � ¬c). Therefore, b � a = 1 and b � ¬c = 1. The latter equality, by (I4), yields b ≤ ¬c. Since b �= 0, we must have b � c. Thus, we have found b ∈ B such that b ∧ (b � a) = b � c. This shows that (B, �) |= (Π7).

(⇐) Suppose (B, �) |= (Π7). Let a �= 0 be an element of B. By (Π7), there is b ∈ B such that b ∧ (b � a) � 0. Therefore, b �= 0 and b � a = 1. Thus, (B, �) |= (I7). �

We next show that ∀∃-statements can be expressed by means of non-standard rules, which we call Π2-rules. The use of non-standard rules in modal logic is not new. One of the pioneers of this approach was Gabbay [16], who introduced a non-standard rule for irreflexivity. A precursor to this work was Burgess [8] who used such rules in the context of branching time logic. We also refer to [17] for the application of non-standard rules to axiomatize the logic of the real line in the language with the Since and Until modalities, and to [28]for a general completeness result for modal languages that are sufficiently expressive to define the so-called difference modality. Our approach is closest to that of Balbiani et al. [1] who use similar non-standard rules in the context of region-based theories of space.

Definition 6.2 (Π2-rule). A Π2-rule is a rule of the form

(ρ) F (ϕ, p) → χ

G(ϕ) → χ

where F, G are formulas, ϕ is a tuple of formulas, χ is a formula, and p is a tuple of propositional letters which do not occur in ϕ and χ.


To each Π2-rule ρ, we associate the ∀∃-statement

Π(ρ) := ∀x, z(G(x) � z → ∃y : F (x, y) � z

).

Definition 6.3. We say that a strict implication algebra (B, �) validates a Π2-rule ρ, and write (B, �) |= ρ, provided (B, �) satisfies Π(ρ).

Consider the Π2-rules:

(ρ6) (ϕ � p) ∧ (p � ψ) → χ

(ϕ � ψ) → χ;

(ρ7) p ∧ (p � ϕ) → χ

ϕ → χ

It is easy to see that Π(ρ6) = (Π6) and Π(ρ7) = (Π7), so by Lemma 6.1, for each (B, �) ∈ RSub, we have:

(B,�) |= (ρ6) iff (B,�) |= (I6);

(B,�) |= (ρ7) iff (B,�) |= (I7).

Definition 6.4 (Proofs with Π2-rules). Let Σ be a set of Π2-rules. For a set of formulas Γ and a formula ϕ, we say that ϕ is derivable from Γ in SIC using the Π2-rules in Σ, and write Γ �Σ ϕ, provided there is a proof ψ1, . . . , ψn such that ψn = ϕ and each ψi is in Γ, or is an instance of an axiom of SIC, or is obtained either by (MP) or (N) from some previous ψj’s, or there is j < i such that ψi is obtained from ψj by an application of one of the Π2-rules ρ ∈ Σ; that is, ψj = F (ξ, p) → χ and ψi = G(ξ) → χ, where F, G are formulas, ξ is a tuple of formulas, χ is a formula, and p is a tuple of propositional letters not occurring in ξ, χ or any of the formulas from Γ that are used in ψ1, . . . , ψi−1 as assumptions.

We next show that the deduction theorem remains true when proofs also involve Π2-rules.

Lemma 6.5. For any set Γ of formulas and for any formulas ϕ, ψ, we have:

Γ ∪ {ϕ} �Σ ψ ⇔ Γ �Σ �ϕ → ψ.

Proof. (⇐) Same as the corresponding proof of Lemma 4.4.(⇒) The only step that is not covered in the corresponding proof of Lemma 4.4 is the step of applying

some Π2-rule ρ ∈ Σ: Suppose there is j < i such that ψj = F (ξ, p) → χ and ψi = G(ξ) → χ for some formulas F, G, a tuple of formulas ξ, a formula χ, and a tuple p of fresh propositional letters not occurring in any of the formulas involved in the proof. We may assume without loss of generality that p also does not occur in ϕ. If this is not the case, then we can rewrite a proof of ψi from Γ ∪ {ϕ} so that the propositional letters p are replaced with fresh propositional letters. By inductive hypothesis, Γ �Σ �ϕ → (F (ξ, p) → χ). Thus, Γ �Σ F (ξ, p) → (�ϕ → χ). Applying ρ yields Γ �Σ G(ξ) → (�ϕ → χ). From this we conclude that Γ �Σ �ϕ → (G(ξ) → χ), as desired. �

Let S be the system obtained by adding the Π2-rules {ρn | n ∈ N} to SIC. Let also U be the inductive subclass of RSub defined by the ∀∃-statements {Π(ρn) | n ∈ N}. We next show that S is strongly sound and complete with respect to U . The proof is a modification of the standard Lindenbaum construction (see, e.g., [23]). The modification follows a similar pattern to the one given in [1, Lem. 7.10].

Theorem 6.6. Let S = SIC+{ρn | n ∈ N}, let U be the inductive subclass of RSub defined by {Π(ρn) | n ∈ N}, and let V be the variety generated by U . For a set of formulas Γ and a formula ϕ, we have:


(1) Γ �S ϕ ⇔ Γ |=U ϕ.(2) �S ϕ ⇔ |=V ϕ.

Proof. (1) That Γ �S ϕ ⇒ Γ |=U ϕ is a straightforward inductive proof on the length of derivations. We only consider the case of Π2-rules.

Suppose ρ is a Π2-rule in S as defined in Definition 6.2, v is a valuation into (B, �) ∈ SIA satisfying Π(ρ), and v(γ) = 1 for each γ ∈ Γ. For simplicity of notation we will write v(Γ) = 1 when v(γ) = 1 for each γ ∈ Γ. We may assume without loss of generality (by re-enumerating all the propositional letters if need be) that the propositional letters p do not occur in any of the formulas in Γ. If G(v(ξ)) � v(χ), then since (B, �) satisfies Π(ρ), there is a tuple c in B such that F (v(ξ), c) � v(χ). Consider the valuation v′

which coincides with v everywhere, except maps p to c. Then v′(F (ξ, p)) = F (v(ξ), c) � v(χ) = v′(χ), so v′(F (ξ, p) → χ) �= 1. Since v′ coincides with v on all propositional letters except p and since we assumed that p do not occur in Γ, we have v′(Γ) = v(Γ) = 1. So we have found a valuation v′ such that v′(Γ) = 1and v′(F (ξ, p) → χ) �= 1, contradicting the inductive hypothesis.

To complete the proof of (1), it remains to show that Γ �S ϕ ⇒ Γ �|=U ϕ. We do this by slightly modifying the construction of the Lindenbaum algebra. Suppose Γ �S ϕ. For each rule ρi, we add a countably infinite set of fresh propositional letters to the set of existing propositional letters. Then we build the Lindenbaum algebra (B, �) over the expanded set of propositional letters, where the elements are the equivalence classes [ϕ] under provable equivalence in S. Next we construct a maximal �-filter M of (B, �) such that {[ψ] | ψ ∈ Γ} ∪ {[¬�ϕ]} ⊆ M and for every rule ρi and formulas ϕ, χ:

(†) if [Gi(ϕ) → χ] /∈ M , then there is a tuple p such that [Fi(ϕ, p) → χ] /∈ M .

To construct M , let A0 := Γ ∪{¬�ϕ}, a consistent set. We enumerate all formulas ϕ as (ϕk : k ∈ N) and all tuples (i, ϕ, χ) where i ∈ N and ϕ, χ are as in the particular rule ρi, and we build the sets A0 ⊆ A1 ⊆· · · ⊆ An ⊆ . . . as follows:

• For n = 2k, if An �S �ϕk, let An+1 = An ∪ {¬�ϕk}; otherwise let An+1 = An.• For n = 2k + 1, let (l, ϕ, χ) be the k-th tuple. If An �S Gl(ϕ) → χ, let An+1 = An ∪ {¬(Fl(ϕ, p) → χ)},

where p is a tuple of propositional letters for ρl not occurring in ϕ, χ, and any of ψ with ψ ∈ An (we can take p from the countably infinite additional propositional letters which we have reserved for the rule ρl). Otherwise, let An+1 = An.

Let A =⋃

n∈N An, SA = {ψ | A �S ψ}, and M = {[ψ] | ψ ∈ SA}. It is easy to see that SA �S ϕ implies ϕ ∈ SA, so M is a filter. Also, for any Γ we have Γ �S ϕ ⇒ Γ �S �ϕ. Therefore, M is a �-filter. Moreover, all An are consistent, and hence so is SA. This implies that M is a proper �-filter. Thus, by the even steps of the construction of the sets An, and by Lemma 4.10(1), M is a maximal �-filter.

Because A ⊆ SA, we have {[ψ] | ψ ∈ Γ} ∪{[¬�ϕ]} ⊆ M . Finally, the odd steps of the construction of the sets An ensure that M satisfies (†). Therefore, we can conclude that M satisfies all the desired properties.

By (†), the quotient of (B, �) by M satisfies each Π(ρi). By Lemma 4.10(1), the quotient belongs toRSub. Therefore, the quotient belongs to U . Moreover, since [¬�ϕ] ∈ M , we have that [¬�ϕ] maps to 1, so [�ϕ] maps to 0 in the quotient. Thus, [ϕ] does not map to 1 in the quotient, while Γ does, and hence Γ �|=U ϕ.

(2) Observe that U consists of the subdirectly irreducible members of V, and apply (1). �The class of subdirectly irreducible algebras in SIA validating a set of Π2-rules is an elementary inductive

subclass of RSub. We next show that the converse is also true. Namely, for every elementary inductive subclass U of RSub, there is a set of Π2-rules {ρi | i ∈ I} such that S = SIC + {ρi | i ∈ I} is strongly sound and complete with respect to U . To obtain such a set of Π2-rules, it is sufficient to show that every


Π2-statement is equivalent to a statement of the form Π(ρ) for some Π2-rule ρ. Without loss of generality we may assume that all atomic formulas Φ(x, y) are of the form t(x, y) = 1 for some term t.

Definition 6.7. Given a quantifier-free first-order formula Φ(x, y), we associate with the tuples of variables x, y the tuples of propositional letters p, q, and define the formula Φ∗(p, q) in the language of SIC as follows:

(t(x, y) = 1)∗ = �t(p, q)

(¬Ψ)∗(x, y) = ¬Ψ∗(p, q)

(Ψ1(x, y) ∧ Ψ2(x, y))∗ = Ψ∗1(p, q) ∧ Ψ∗

2(p, q)

Lemma 6.8. Let (B, �) ∈ RSub and Φ(x, y) be a quantifier-free formula.

(1) (B, �) satisfies Φ(x, y) iff (B, �) satisfies the formula Φ∗(p, q).(2) (B, �) satisfies ∀x∃yΦ(x, y) iff (B, �) satisfies ∀x, z

(1 � z → ∃y : Φ∗(x, y) � z

).

Proof. (1) For each term t(x, y), we evaluate x, p as a and y, q as b, where a, b are tuples of elements of B. It is obvious that t(a, b) = 1 implies �t(a, b) = 1, and t(a, b) �= 1 implies �t(a, b) = 0. This shows the equivalence for atomic formulas, and an easy induction then proves it for all quantifier-free formulas.

(2) (⇒) Suppose (B, �) |= ∀x∃yΦ(x, y). Let a be a tuple of elements of B and c ∈ B. By assumption, there exists a tuple b in B such that (B, �) |= Φ(x, y)[a, b]. Therefore, by (1), if 1 � c, then Φ∗(a, b) = 1 � c. Thus, (B, �) |= ∀x, z

(1 � z → ∃y : Φ∗(x, y) � z

).

(⇐) Suppose (B, �) |= ∀x, z(1 � z → ∃y : Φ∗(x, y) � z

). Let a be a tuple of elements of B. Since 1 � 0,

there exists a tuple b in B such that Φ∗(a, b) � 0. Therefore, since Φ∗(a, b) evaluates only to 0 or 1, we obtain Φ∗(a, b) = 1. Thus, by (1), (B, �) |= Φ(x, y)[a, b]. This shows that (B, �) |= ∀x∃yΦ(x, y). �

Consequently, an arbitrary Π2-statement ∀x∃yΦ(x, y) is equivalent to the Π2-statement associated to the Π2-rule

(ρΦ) Φ∗(ϕ, p) → χ

χ

Thus, by Theorem 6.6, we obtain:

Theorem 6.9. If T is a Π2-theory of first-order logic axiomatizing an inductive subclass U of RSub, then the system S = SIC + {ρΦ | Φ ∈ T} is strongly sound and complete with respect to U ; that is, for a set of formulas Γ and a formula ϕ, we have:

Γ �S ϕ ⇔ Γ |=U ϕ.

6.2. Admissibility of Π2-rules

By Theorem 5.8, S2IC is strongly sound and complete with respect to Com. On the other hand, it follows from Theorem 6.9 that S2IC together with (ρ6) and (ρ7) is also strongly sound and complete with respect to Com. Therefore, the rules (ρ6) and (ρ7) are admissible in S2IC. We next give a general criterion of admissibility for Π2-rules in SIC and S2IC. This yields an alternative proof that (ρ6) and (ρ7) are admissible in S2IC.

Definition 6.10. A rule ρ is admissible in a system S if for each formula ϕ, from �S+ρ ϕ it follows that �S ϕ.


Lemma 6.11. If a Π2-rule

(ρ) F (ϕ, p) → χ

G(ϕ) → χ

is admissible in S ⊇ SIC, then Γ �S+ρ ϕ ⇔ Γ �S ϕ.

Proof. It suffices to show that for any set of formulas Γ and any tuple ϕ, χ of formulas, if Γ �SIC F (ϕ, p) → χ

and p does not occur in Γ, ϕ, χ, then Γ �SIC G(ϕ) → χ.Suppose Γ �SIC F (ϕ, p) → χ and p does not occur in Γ, ϕ, χ. Then there is a finite Γ0 ⊆ Γ such that

Γ0 �SIC F (ϕ, p) → χ. Let ψ =∧

Γ0, so {ψ} �SIC F (ϕ, p) → χ. By Theorem 4.4, �SIC �ψ → (F (ϕ, p) → χ), so �SIC F (ϕ, p) → (�ψ → χ). Since p does not occur in ϕ, �ψ → χ, by admissibility of ρ, we have �SIC G(ϕ) → (�ψ → χ). Therefore, �SIC �ψ → (G(ϕ) → χ), and applying Theorem 4.4 again yields {ψ} �SIC G(ϕ) → χ. Thus, Γ �SIC G(ϕ) → χ. �Theorem 6.12 (Admissibility Criterion).

(1) A Π2-rule ρ is admissible in SIC iff for each (B, �) ∈ RSub there is (C, �) ∈ RSub such that (B, �)is isomorphic to a substructure of (C, �) and (C, �) |= Π(ρ).

(2) A Π2-rule ρ is admissible in S2IC iff for each (B, �) ∈ Con there is (C, �) ∈ Con such that (B, �) is isomorphic to a substructure of (C, �) and (C, �) |= Π(ρ).

Proof. (1) (⇒) Suppose ρ is admissible in SIC. It is sufficient to show that there exists a model (C, �) of the theory

T = Th(RSub) ∪ {Π(ρ)} ∪ Δ(B,�)

where Δ(B, �) is the diagram of (B, �) [11, p. 68]. Suppose for a contradiction that T has no models, hence is inconsistent. Then, by compactness, there exists a quantifier-free first-order formula Ψ(x) and a tuple pa of propositional letters corresponding to a ∈ B such that

Th(RSub) ∪ {Π(ρ)} |= ¬Ψ(pa) and (B,�) |= Ψ(a).

We enrich the language of SIC+ρ with {pa}. By Theorem 6.6, SIC+ρ is complete with respect to the algebras in RSub satisfying Π(ρ). Therefore, by Lemma 6.8(1), �SIC+ρ (¬Ψ(pa))∗ where (−)∗ is the translation given in Definition 6.7. By admissibility, �SIC (¬Ψ(pa))∗. Thus, for each valuation v into B that maps pa to a, we have v((¬Ψ(pa))∗) = 1, so v((Ψ(pa))∗) = 0. This contradicts the fact that (B, �) |= Ψ(a). Consequently, Tmust be consistent, and hence it has a model.

(⇐) To prove that ρ is admissible in SIC it is sufficient to show that if there is a proof of F (ϕ, p) → χ, then there is a proof of G(ϕ) → χ. Suppose �SIC F (ϕ, p) → χ with p not occurring in ϕ, χ. Assume (B, �) ∈ RSub and let v be a valuation on (B, �). By assumption, there is (C, �) ∈ RSub such that (B, �) is isomorphic to a substructure of (C, �) and (C, �) |= Π(ρ). Let i : B → C be the embedding. Then v′ := i ◦v is a valuation on (C, �). For any c ∈ C, let v′′ be the valuation (v′)cp. Since �SIC F (ϕ, p) → χ, by Theorem 4.11, v′′(F (ϕ, p) → χ) = 1C . This means that for all c ∈ C, we have F (v′(ϕ), c) ≤ v′(χ). Therefore, (C, �) |= ∀y

(F (v′(ϕ), y) ≤ v′(χ)

). Since (C,�) |= Π(ρ), we have (C, �) |= G(v′(ϕ)) ≤ v′(χ).

Thus, as G(v′(ϕ)) ≤ v′(χ) in C, we have G(v(ϕ)) ≤ v(χ) in B. Consequently, v(G(ϕ) → χ) = 1B . Applying Theorem 4.11 again yields that �SIC G(ϕ) → χ, and hence ρ is admissible in SIC.

(2) The proof is similar to that of (1) and uses the fact that S2IC is strongly sound and complete with respect to Con. �


Corollary 6.13.

(1) (ρ6) is admissible in SIC and S2IC.(2) (ρ7) is admissible in SIC and S2IC.

Proof. (1) For admissibility of (ρ6) in SIC apply Theorem 6.12(1) and Lemmas 6.1(1) and 5.4(1). For admissibility of (ρ6) in S2IC, apply Theorem 6.12(2) and Lemmas 6.1(1) and 5.4(2).

(2) For admissibility of (ρ7) in SIC apply Theorem 6.12(1) and Lemmas 6.1(2) and 5.6. For admissibility of (ρ7) in S2IC apply Theorem 6.12(2) and Lemmas 6.1(2) and 5.6. �6.3. Calculi for zero-dimensional and connected compact Hausdorff spaces

In the remainder of this section, we will consider zero-dimensional and connected compact Hausdorff models, and identify their logics. Starting with zero-dimensionality, we consider the following property, studied in [4]:(S9) a ≺ b implies ∃c : c ≺ c and a ≺ c ≺ b.

The corresponding ∀∃-statement is(Π9) ∀x, y, z

(x � y � z → ∃u : (u � u) ∧ (x � u) ∧ (u � y) � z

).

Lemma 6.14. Let (B, �) ∈ Com. Then (B, �) |= (S9) iff (B, �) |= (Π9).

Proof. (⇒) Suppose a � b � d. Then d �= 1 and a � b �= 0, so a � b = 1. Therefore, a ≺ b, and so by (S9), there is c such that c ≺ c and a ≺ c ≺ b. Thus, (c � c) ∧ (a � c) ∧ (c � b) = 1 � d. Consequently, (B, �) |= (Π9).

(⇐) Suppose a ≺ b. Then a � b = 1 � 0. Therefore, by (Π9), there is c such that (c � c) ∧ (a �c) ∧ (c � b) � 0, which implies (c � c) ∧ (a � c) ∧ (c � b) = 1. Thus, c ≺ c and a ≺ c ≺ b. Consequently, (B, �) |= (S9). �

The Π2-rule corresponding to (Π9) is

(ρ9) (p � p) ∧ (ϕ � p) ∧ (p � ψ) → χ

(ϕ � ψ) → χ

Theorem 6.15. (ρ9) is admissible in SIC and S2IC.

Proof. It is easy to see that the (C, ≺) constructed in the proof of Lemma 5.4 satisfies (S9). Therefore, for admissibility in SIC, we can apply Theorem 6.12(1), Lemma 6.14, and Lemma 5.4(1); and for admissibility in S2IC, we can apply Theorem 6.12(2), Lemma 6.14, and Lemma 5.4(2). �

As a consequence of Theorem 6.15 we obtain:

Corollary 6.16. S2IC is strongly sound and complete with respect to the class of compingent algebras satisfying(S9).

Following [4, Def. 4.5], we call a de Vries algebra zero-dimensional if it satisfies (S9), and denote the class of zero-dimensional de Vries algebras by zDeV. Let (B, �) ∈ Com satisfy (S9), and let X be the de Vries dual of (B, �). It follows from de Vries duality and [4, Lem. 4.11] that X is zero-dimensional, and hence (RO(X), �) is a zero-dimensional de Vries algebra. Thus, each (B, �) ∈ Com satisfying (S9) embeds in a


zero-dimensional de Vries algebra. We recall from Section 2 that zero-dimensional compact Hausdorff spaces are called Stone spaces, and denote the class of Stone spaces by Stone. As a consequence of Corollary 6.16we then have:

Theorem 6.17.

(1) S2IC is strongly sound and complete with respect to zDeV.(2) S2IC is strongly sound and complete with respect to Stone.

Turning to connectedness, consider the following property:(S10) a ≺ a implies a = 0 or a = 1.

Clearly (B, �) ∈ Com satisfies (S10) iff a � a ≤ �a ∨ �¬a holds in (B, �). Therefore, (B, �) satisfies (S10) iff (B, �) |= (C), where (C) is the formula(C) (ϕ � ϕ) → (�ϕ ∨ �¬ϕ).

Definition 6.18. The connected symmetric strict implication calculus CS2IC is the extension of S2IC with the axiom (C).

We call (A, �) ∈ S2IA connected if (A, �) satisfies a � a ≤ �a ∨ �¬a for each a ∈ A. Let CS2IA be the subvariety of S2IA consisting of connected symmetric strict implication algebras. We also call a compingent algebra connected if it satisfies (S10), and denote the class of connected compingent algebras by CCom. As a simple consequence of Theorems 5.2 and 6.6 we have:

Corollary 6.19.

(1) CS2IC is strongly sound and complete with respect to CS2IA.(2) CS2IC is strongly sound and complete with respect to CCom.

Lemma 6.20. A compingent algebra (B, ≺) satisfies (S10) iff its dual compact Hausdorff space X is connected.

Proof. If X is connected, then ∅, X are the only clopen subsets of X. Therefore, for U ∈ RO(X), we have U ≺ U implies U = ∅ or U = X. Thus, (RO(X), ≺) satisfies (S10). Since (B, ≺) is isomorphic to a subalgebra of (RO(X), ≺), we conclude that (B, ≺) satisfies (S10).

Conversely, let U be clopen in X. Then U =⋃{β(a) : β(a) ⊆ U} and the family {β(a) : β(a) ⊆ U} is

up-directed. Because U is compact, there is a ∈ B such that β(a) = U . As β : B → RO(X) is an embedding and (B, ≺) satisfies (S10), U = β(0) = ∅ or U = β(1) = X. Thus, X is connected. �

As an immediate consequence we obtain:

Lemma 6.21. A de Vries algebra (B, ≺) satisfies (S10) iff its de Vries dual X is connected.

We call a de Vries algebra connected if it satisfies (S10), and denote the class of connected de Vries algebras by cDeV. By Lemma 6.21, each (B, ≺) ∈ CCom embeds in a connected de Vries algebra. Let cKHaus be the class of connected compact Hausdorff spaces. Then Corollary 6.19 and Lemma 6.21 imply:

Theorem 6.22.

(1) CS2IC is strongly sound and complete with respect to cDeV.


(2) CS2IC is strongly sound and complete with respect to cKHaus.

The table below summarizes our completeness results.

Logic Complete with respect toSIC SIA; RSubS2IC S2IA; Con; Com; DeV; zDeV

KHaus; StoneCS2IC CS2IA; CCom; cDeV; cKHaus

Remark 6.23. By Theorems 5.2 and 5.10, S2IC is strongly sound and complete with respect to Con and DeV. Thus, the logic of contact algebras is the same as the logic of de Vries algebras. On the other hand, the Π2-theories (the sets of valid Π2-rules) of Con and DeV are obviously different as the Π2-rules (ρ6) and (ρ7) belong to the latter, and hence to the theory of compact Hausdorff spaces, but not to the former. This generates an interesting methodological question of what the right logical formalism should be to reason about compact Hausdorff spaces. Should we be concerned only with the logics or should we also consider the theories of Π2-rules? Although in this paper we are only concerned with logics, our results suggest that a theory of Π2-rules may be a more appropriate framework to reason about compact Hausdorff spaces. We leave it as a future work to develop the Π2-theory for compact Hausdorff spaces together with the general theory of such calculi.

7. Comparison with relevant work

In this final section we compare our approach to that of Balbiani et al. [1]. Namely, we show how to translate fully and faithfully the language L(C, ≤) of [1] into our language. We start by recalling the concept of contact relation, one of the key concepts of region-based theory of space; see, e.g., [27]. A binary relation C on a Boolean algebra B is a precontact relation if it satisfies:

(C1) aCb ⇒ a, b �= 0.(C2) aC(b ∨ c) ⇔ aCb or aCc.(C3) (a ∨ b)Cc ⇔ aCc or bCc.

A precontact relation is a contact relation if it satisfies:

(C4) a �= 0 implies aCa.(C5) aCb implies bCa.

As was pointed out in [6, Rem. 2.5] (cf. Remark 3.2), there is a one-to-one correspondence between subor-dinations and precontact relations. Namely, if ≺ is a subordination, then the relation C≺ defined by aC≺b

iff a ⊀ ¬b is a precontact relation; if C is a precontact relation, then the relation ≺C defined by a ≺C b iff a/C¬b is a subordination; and this correspondence is one-to-one. Moreover, a subordination ≺ satisfies (S5) iff the corresponding precontact relation C≺ satisfies (C4), and ≺ satisfies (S6) iff C≺ satisfies (C5). Thus, contact relations are in one-to-one correspondence with subordinations satisfying (S5) and (S6).

On regular open sets of a compact Hausdorff space X the contact relation is defined by UCV iff Cl(U) ∩Cl(V ) �= ∅. If R is a reflexive and symmetric relation on a set X, then the contact relation CR is defined on P(X) by UCRV iff R[U ] ∩ V �= ∅.

We next recall that the formulas of the language L(C, ≤) are built from atomic formulas using Boolean connectives ¬, ∧, ∨, →, ⊥, �; atomic formulas are of the form tCs and t ≤ s, where t, s are Boolean terms


(C stands for the contact relation and ≤ for the inclusion relation). In turn, Boolean terms are built from Boolean variables using Boolean operations �, �, (−)∗, 0, 1.

As usual, a Kripke frame is a pair (W, R), where W is a nonempty set and R is a binary relation on W , and a valuation is a map v from the set of Boolean variables to the powerset P(W ). It extends to the set of all Boolean terms as follows:

v(t � s) = v(t) ∩ v(s),

v(t � s) = v(t) ∪ v(s),

v(t∗) = W \ v(t),

v(0) = ∅,

v(1) = W.

A Kripke model is a triple (W, R, v) consisting of a Kripke frame (W, R) and a valuation v. Atomic formulas are interpreted in (W, R, v) as follows:

(W,R, v) |= (t ≤ s) ⇔ v(t) ⊆ v(s),

(W,R, v) |= (tCs) ⇔ R[v(t)] ∩ v(s) �= ∅.

Complex formulas are then interpreted by the induction clauses for propositional connectives.In [1, Sec. 6] the authors define the propositional calculus PWRCC in the language L(C, ≤) and prove

that PWRCC is sound and complete with respect to the class of Kripke frames where the binary relation R is reflexive and symmetric. Such Kripke frames are closely related to contact algebras. Namely, as we already pointed out in Sections 2 and 5, the following lemma holds.

Lemma 7.1.

(1) Suppose (W, R) is a reflexive and symmetric Kripke frame. Define ≺R on P(W ) by U ≺R V iff R[U ] ⊆V . Then (P(W ), ≺R) is a contact algebra.

(2) Suppose (B, ≺) is a contact algebra and (X, R) is the dual of (B, ≺). Then (X, R) is a reflexive and symmetric Kripke frame, and the Stone map β : B → P(X), given by β(a) = {x ∈ X | a ∈ x}, is an embedding of (B, ≺) into (P(X), ≺R).

We next translate L(C, ≤) into our language L. We identify the set of Boolean variables of L(C, ≤) with the set of propositional letters of L. Then Boolean terms can be translated into formulas of L as follows:

aT = a, for a Boolean variable a,

(t � s)T = tT ∧ sT ,

(t � s)T = tT ∨ sT ,

(t∗)T = ¬(tT ),

0T = ⊥,

1T = �.

For atomic formulas, we define:


(t ≤ s)T = �(tT → sT ),

(tCs)T = ¬(tT � ¬sT ).

Finally, complex formulas are translated inductively as follows:

(¬ϕ)T = ¬ϕT ,

(ϕ ∧ ψ)T = ϕT ∧ ψT ,

(ϕ ∨ ψ)T = ϕT ∨ ψT ,

(ϕ → ψ)T = ϕT → ψT ,

⊥T = ⊥,

�T = �.

Theorem 7.2. For any formula ϕ of L(C, ≤), we have

PWRCC � ϕ iff S2IC � ϕT .

Proof. By [1, Cor. 6.1], PWRCC is sound and complete with respect to the class of reflexive and symmetric Kripke frames (W, R); and by Theorem 5.2, S2IC is sound and complete with respect to the class of contact algebras. Given a Kripke model (W, R, v), the valuation v of Boolean variables of L(C, ≤) into P(W ) can be seen as a valuation of propositional letters of L into the algebra (P(W ), �R).

Claim. (W, R, v) |= ϕ iff (P(W ), �R, v) |= ϕT .

Proof of Claim. For a Boolean term t, we have v(t) = v(tT ) ⊆ W . If ϕ is an atomic formula of the form t ≤ s, then

(W,R, v) |= ϕ iff v(t) ⊆ v(s)

iff v(tT ) ≤ v(sT ) in P(W )

iff (P(W ),�R, v) |= tT → sT

iff (P(W ),�R, v) |= �(tT → sT )

iff (P(W ),�R, v) |= ϕT .

If ϕ is an atomic formula of the form tCs, then

(W,R, v) |= ϕ iff R[v(t)] ∩ v(s) �= ∅

iff R[v(t)] � W \ v(s)iff R[v(tT )] � v(¬sT )

iff (P(W ),�R, v) |= ¬(tT � ¬sT )

iff (P(W ),�R, v) |= ϕT .

Finally, if ϕ is a complex formula, then a straightforward induction completes the proof. �Now, if PWRCC � ϕ, then there is a reflexive and symmetric Kripke model (W, R, v) refuting ϕ. By

the Claim, ϕT is refuted in (P(W ), �R, v). Therefore, S2IC � ϕT . Conversely, if S2IC � ϕT , then there


is a contact algebra (B, ≺) and a valuation v on (B, ≺) refuting ϕT . By Lemma 7.1(2), ϕT is refuted in (P(X), �R, v). By the Claim, ϕ is refuted in (X, R, v). Thus, PWRCC � ϕ. �

As was pointed out to us by D. Vakarelov, our language, like the language of [1], admits a translation tr into the basic language of modal logic enriched with the universal modality. We conclude the paper by spelling out this connection.

We recall that K denotes the basic modal logic, KT := K + (�p → p), KTB := KT + (p → ��p), and S5 := KTB + (�p → ��p) (see, e.g., [7, Sec. 4.1]). We also recall that the universal modality [∀] is an S5-modality satisfying [∀]p → �p. Let KTU, KTBU, and S5U be the extensions of KT, KTB, and S5 with the universal modality (see, e.g., [7, Sec. 7.1]).

It is well known that all the above logics are Kripke complete and have the finite model property (see, e.g., [7]). In particular, it is known that KTU has the finite model property with respect to the finite reflexive Kripke frames enriched with the universal relation, that KTBU has the finite model property with respect to the finite reflexive and symmetric frames enriched with the universal relation, and that S5U has the finite model property with respect to the finite reflexive, symmetric, and transitive frames enriched with the universal relation (see, e.g., [20]).

Consider the following translation tr from our language into the modal language enriched with the universal modality:

tr(p) = p,

tr(¬ϕ) = ¬tr(ϕ),

tr(ϕ ∧ ψ) = tr(ϕ) ∧ tr(ψ),

tr(ϕ � ψ) = [∀](tr(ϕ) → �tr(ψ)).

Theorem 7.3. Let ϕ be a formula of L.

(1) SIC � ϕ iff KTU � tr(ϕ).(2) S2IC � ϕ iff KTBU � tr(ϕ).(3) S2IC � ϕ iff S5U � tr(ϕ).

Proof. (1) Let (B, ≺) ∈ RSub and let (X, R) be the dual of (B, ≺). By Lemma 2.2(1), R is reflexive. Let v be a valuation on Clop(X). By defining the universal relation on X, we can view (X, R, v) as a model of KTU. We prove that for each formula ψ and x ∈ X,

x ∈ v(ψ) iff (X,R, v), x |= tr(ψ). (1)

This can be done by induction on the complexity of ψ, and the only nontrivial case is ψ = χ � ξ. If x /∈ v(χ � ξ), then R[v(χ)] � v(ξ). So there exist y, z ∈ X such that y ∈ v(χ), yRz and z /∈ v(ξ). By the induction hypothesis, (X, R, v), y |= tr(χ) and (X, R, v), z �|= tr(ξ). Thus, (X, R, v), y �|= tr(χ) → �tr(ξ), and so (X, R, v), x �|= [∀](tr(χ) → �tr(ξ)). The converse implication is proved similarly. From (1) we obtain that

(Clop(X),�R, v) |= ψ iff (X,R, v) |= tr(ψ). (2)

Now, if SIC � ϕ, then by Theorem 4.11, there is (B, ≺) ∈ RSub such that (B, �) refutes ϕ. Therefore, there is a valuation v on Clop(X) such that (Clop(X), �R, v) �|= ϕ. By (2), (X, R, v) �|= tr(ϕ), and hence KTU � tr(ϕ). Conversely, if KTU � tr(ϕ), then there is a finite reflexive model (X, R, v) with the universal


relation such that (X, R, v) �|= tr(ϕ). Since (X, R) is finite, we may view it as the dual of (P(X), ≺R). By (2), (X, R, v) �|= tr(ϕ) implies (P(X), �R, v) �|= ϕ. Since (P(X), ≺R) ∈ RSub, we conclude that SIC � ϕ.

The proof of (2) is similar to (1) but uses Theorem 5.2.The proof of (3) is similar to (1) and (2) but uses Theorem 5.8. �The theorem above also indicates that unlike the classical modal case, where reflexivity, symmetry,

and transitivity are all expressible, in our system we can only express reflexivity and symmetry, but not transitivity as S2IC is complete with respect to (X, R) where R is an equivalence relation. Thus, our language cannot distinguish between KTBU and S5U. This is yet another motivation to investigate non-standard rules and inductive classes of subordination algebras further.

Declaration of Competing Interest

There is no competing interest.

Acknowledgements

We are extremely thankful to the referee for careful reading and substantial comments. This has resulted in a completely revamped and restructured paper, with streamlined proofs, which is much more readable. In addition, the referee has spotted a large number of inaccuracies, which have been corrected. We also thank Philippe Balbiani and Dimiter Vakarelov for useful comments.

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